Ecole doctorale Ondes et Matière
thèse
pour l’obtention du grade de Docteur en sciences de l’Université Paris XI présentée par
Fred JENDRZEJEWSKI
Quantum transport of ultracold atoms in disordered potentials:
Anderson Localization in three dimensions Coherent Backscattering
thèse soutenue le 6 Novembre 2012 devant le jury composé de : M. Frédéric Chevy
M. Robin Kaiser M. Pierre Pillet M. Immanuel Bloch M. Philippe Bouyer M. Vincent Josse M. Alain Aspect
M. Laurent Sanchez-Palencia
Rapporteur Rapporteur Examinateur Examinateur Directeur de thèse Examinateur Membre invité Membre invité
There are so many people that made the last three years possible that it will be impossible for me to give them all the attention they merit. So, I will start it was a huge thank you to everyone I worked with during the last three years. Thanks to their support, I was introduced into the scientific way of life and it was a blast.
The CNRS was kind enough to support my PhD work financially, which allowed me to get my feet wet in the field of fundamental research. I had to great pleasure to make this first experience in the very professional environment of the Atom Optics group at the Laboratoire Fabry at the Institut Optique. The group leaders, Alain Aspect and Patrick Bouyer, have built up a group, which breathes the passion for experimental physics.
During this time I learned a lot, most importantly that I am probably wrong whenever I think that I understood my experiments "well". At the beginning of my PhD I was naive enough to think that things would work the way they are supposed to work. In the end they did, but I had no idea of how to difficult it is to get our experiments working properly.
Luckily, I had a great team of teachers in Alain Bernard and Patrick Cheinet that thought me a lot of the things I know about now. Patricks love for details and Alains’ creative solutions for fixing things, which often enough involved some paper and scotch tape, are attributes I really learned to appreciate.
And if even Alain and Patrick would have some doubts, we had our team leader Vincent Josse. Vincent guided the evolution of this experiment and my studies with a lot of personal investment. Vincent has a lot of the properties, I now think are essential for being a great physicist. He teached me the patience it takes to get the experiment running properly and to find a clean signature of the things that were going on. His impressive ability to explain complicated things in an intuitive way, has helped me a lot in order to understand what we were doing. This drive to explain things in a pedagogical way also made it a pleasure to write down to the results we had obtained. Alain Aspect and Vincent, teached us younger team members how to write clear papers.
Unfortunatly, Alain and Patrick had to leave the team, when the most exiting results started come in. It was a pleasure to have Kilian joining our team. He was a great partner, who never lost his calm and kindness, even after months of problems. But the team "Pince" is only a tiny part of the big Atom Optics group with all its great permanent members and fun to be with PhD students.
Additionally to my first experience in research, I teached for my first time at the Institut d’Optique. I thank all my collaborators from the Polarization and Electronics team for the great team spirit and their patience with me.
Finally I have to thank my family and friends. Given the energy it takes to do a PhD, I would not have been able to go through with it without the help of my wife, my family and my friends. They allowed me get the necessary distance after some intense weeks of work and come back with fresh enthusiasm.
1 Introduction 9
2 Wave propagation in disorder 15
2.1 Interference and disorder . . . 15
2.1.1 Interferences in electron transport in disordered media . . . 16
2.2 From weak localization... . . 17
2.2.1 Experiments on weak localization and backscattering . . . 19
2.3 ... to the suppression of transport - Anderson Localization . . . 20
2.3.1 Experiments on Anderson Localization . . . 22
2.4 Experiments with ultracold atoms . . . 23
2.5 Conclusion - Towards higher dimensions and interacting bosons . . . 24
3 Production of coherent matter waves 25 3.1 Generalities about Bose-Einstein condensates . . . 25
3.1.1 Bose-Einstein condensation and the Gross-Pitaevskii Equation . . . 25
3.1.2 The Thomas-Fermi Regime . . . 26
3.1.3 Expansion of a BEC: time-of-flight experiments . . . 27
3.1.3.1 The scaling approach . . . 28
3.1.3.2 Ballistic expansion . . . 28
3.1.3.3 Expansion from an isotropic trap . . . 29
3.2 Manipulation of the atoms . . . 29
3.2.1 Magnetic potentials . . . 29
3.2.2 Optical potentials . . . 29
3.2.2.1 The gaussian beam . . . 31
3.3 The preparation of a cloud of ultracold atoms . . . 32
3.3.1 Imaging . . . 33
3.3.1.1 Absorption in the cooling chamber . . . 34
3.3.2 Cooling . . . 34
3.3.3 Transport with an optical tweezer . . . 35
3.4 The second chamber . . . 36
3.4.1 Imaging in the second chamber . . . 38
3.4.2 Evaporation in an optical dipole trap . . . 40
3.4.2.1 The crossed dipole trap . . . 41
3.4.2.2 The evaporation scheme . . . 42
3.4.2.3 Characteristics of the cloud during the evaporation . . . 44
3.4.3 The magnetic levitation . . . 44
3.4.3.1 A model of our levitation . . . 46
3.4.3.2 Experimental implementation . . . 48
3.4.4 Expansion of the BEC . . . 49
3.5 Conclusion . . . 51
4 Characteristics of a three-dimensional speckle disorder 53 4.1 The speckle - a well-controlled, correlated disorder . . . 53
4.1.1 Simple model of the speckle as an diffraction picture . . . 54
4.1.2 Physical model of the speckle . . . 55
4.1.3 Intensity distribution . . . 55
4.1.4 Spatial properties and dimensions of a speckle grain . . . 56
4.1.4.1 Auto-correlation function . . . 56
4.1.4.2 Transverse direction . . . 57
4.1.4.3 Longitudinal direction . . . 57
4.1.4.4 The power spectral density . . . 58
4.2 Realization of a three-dimensional speckle disorder . . . 58
4.2.1 Intensity distribution . . . 59
4.2.2 Correlation function . . . 60
4.2.3 Experimental setup . . . 61
4.3 Conclusion . . . 62
5 Some notions of diffusion and localization of matter waves 65 5.1 From diffusion in a weak disorder... . . 65
5.1.1 Diffusion as a random walk in a disordered potential . . . 66
5.2 ...to localization in a strong disorder . . . 67
5.2.1 Weak localization . . . 67
5.2.2 The Thouless criterion . . . 68
5.3 Scaling theory . . . 70
5.3.1 Consequences of scaling theory - Localization of all states in one and two dimensions . . . 71
5.3.2 The existence of a mobility edge in 3D . . . 72
5.3.3 The critical region . . . 73
5.4 Microscopic quantaties - Energy scales . . . 73
5.4.1 The scattering time . . . 74
5.4.2 The energy spread and the weak disorder condition . . . 75
5.4.3 The transport length . . . 76
5.4.4 The correlation energy - from classical to quantum disorder . . . 77
5.4.5 The mobility edge in the quantum disorder regime . . . 77
5.4.6 The mobility edge in the classical disorder regime . . . 78
5.4.7 The classical percolation threshold . . . 79
5.4.8 Percolation in an anisotropic 3D speckle . . . 80
5.4.9 The self-consistent theory of localization . . . 81
5.5 Summary - a generic phase diagram in 3D . . . 83
6 Experiments on 3D Localization 85 6.1 Localization in a weak disorder in 1D . . . 85
6.1.1 Key features 1D localization in a speckle disorder . . . 86
6.1.2 Experimental observations . . . 87
6.2 Transposing the 1D scheme to 3D . . . 88
6.2.1 The quantum regime . . . 88
6.2.2 Persistence of the diffusive component at arbitrarily strong disorder 89 6.2.3 Properties of the diffusive component . . . 90
6.2.4 Properties of the localized component . . . 91
6.2.5 Intermediate summary . . . 91
6.3 Expansion of ultra-cold atoms in a strong speckle disorder in 3D . . . 92
6.3.1 The experimental setup . . . 92
6.3.2 Observations . . . 94
6.3.3 Phenomenological model . . . 95
6.3.4 The diffusive component . . . 95
6.3.5 The localized component . . . 97
6.3.6 Trapped fraction . . . 98
6.3.7 Summary of the experimental observations . . . 99
6.4 Comparision to the theoretical model . . . 100
6.4.1 The localized shape and fraction . . . 100
6.4.2 The diffusive component . . . 102
6.4.3 The profiles . . . 103
6.4.4 Conclusions . . . 103
6.5 First studies on the influence of the disorder properties . . . 104
6.5.1 Expansion in an incoherently crossed speckle . . . 104
6.5.2 Modification of the correlations in the disorder . . . 105
6.6 Towards a better control of the energy distribution . . . 106
6.6.1 Experiments on the momentum distribution . . . 106
6.6.2 Ramped scheme . . . 107
6.7 Conclusions . . . 108
7 Experiments on Coherent Backscattering 111 7.1 The coherent backscattering mechanism . . . 111
7.2 The experimental sequence . . . 113
7.2.1 Delta-kick cooling . . . 114
7.2.2 The 2D configuration in an elongated disorder . . . 115
7.3 Investigations of the momentum distribution . . . 116
7.3.1 The scattering time . . . 118
7.3.2 The transport time . . . 119
7.3.3 Analysis of the CBS peak . . . 119
7.4 Dependence on the microscopic properties . . . 121
7.4.1 The scattering time . . . 121
7.4.2 The transport time . . . 123
7.4.3 The CBS signal . . . 124
7.5 Conclusions . . . 125
8 Conclusion 127
A Atom losses during the expansion in the disorder 129
B Publications 131
Bibliography 132
Introduction
Modern laser cooling [Cohen-Tannoudji 98a, Phillips 98, Chu 98] and evaporative cool- ing made it possible to observe in 1995 [Ketterle 02, Cornell 02] the accumulation of a macroscopic number of alkaline atoms in a single quantum state, called Bose-Einstein condensation [Einstein 27]. Today, these systems provide a very precise control of the experimental parameters [Bloch 08]:
• The control of the trapping potential makes it possible to let the atoms evolve in dif- ferent geometries. This is especially interesting in order to study low-dimensional sys- tems. For example it was possible to observe a gas of bosons in the Tonks-Giradeau regime in one dimension (1D) [Kinoshita 04, Paredes 04] and the Berezinskii-Kosterlitz- Thouless (BKT) transition in two dimensions (2D) [Hadzibabic 06].
• Inter-particle interactions are well controlled by Fano-Feshbach resonances, which can be controlled by magnetic fields [Chin 10].
• The atomic density profile can be directly imaged, either by measuring the absorption of a resonant laser traversing the cloud or by measuring the fluorescence emitted by the illuminated atoms.
Therefore, it is now possible to realize ideal systems, which allow us to simulate complex theoretical problems from Condensed Matter Physics. Two emblematic examples perfectly illustrate this possibility: the realization of the Mott transition in the Bose-Hubbard model, which describes the competition between inter-atomic interactions and hopping in the lattice [Greiner 02], and the observation of the BEC-BCS crossover, see [Greiner 03, Bourdel 04] and references therein.
In this context, the group of Pr. Alain Aspect has pioneered another type of quantum simulators: the study of ultracold atoms in controlled disordered potentials [Clément 05, Sanchez-Palencia 07, Billy 08]. Studying such disordered systems is of utmost interest:
they lie at the heart of many fundamental phenomena, such as Anderson localization in disordered electronic conductors [Anderson 58], superfluidity in porous media [Reppy 92], and possibly high-Tc superconductivity [Goldman 98]. In spite of extensive studies, the understanding of such systems remains an exciting but formidable task, many issues still being unresolved or even controversial (see e.g.[Weichman 08, Aspect 09, Lagendijk 09, Sanchez-Palencia 10, Burmistrov 12]).
When I arrived in the end of 2008, the team had just proven the potential of their experimental setup with the landmark experiments in which they observed the celebrated Anderson Localization (AL) in one dimension [Billy 08, Roati 08] shown in Fig. 1.1. AL is the most emblematic effect of disorder, describing the appearance of localized states due to interference between many scattering amplitudes associated with the diffusion of a single quantum particle. Motivated by the success of those experiments we aimed to
c
atomic density (a.u.)
Position (mm)
Figure 1.1: Anderson Localization in 1D [Billy 08]. a) A very dilute condensate is created in a hybrid trap (The magnetic field is present in grey). b) When the magnetic trap is switched off, the condensate propagates along the optical guide in the presence of a weak disorder and stops after∼1s. c) Profile in a semilogarithmic scale once the stationary localized regime has been reached. The exponential decay, emblematic for AL, is clearly visible.
first refine our understanding of AL in one dimension (1D) and then to transpose the set-up to the higher dimensional case. In fact, AL depends strongly on the dimension of the system [Abrahams 79]. A quantum phase transition around a mobility edge is predicted in three dimensions (3D). This mobility edge corresponds to an energy threshold separating localized from extended states. Determining the value of that mobility edge and exploring the critical regime around it remains a challenge for microscopic theory, numerical simulations, and experiments [Lagendijk 09]. Nowadays, the originally very abstract concept of AL has even gained some importance for possible applications in optical fibers [Karbasi 12] and possibly even the optimization of the efficiency of solar panels [Vynck 12].
Thesis
This thesis was realized in the Atom Optics group, led by Alain Aspect, in Palaiseau. The results presented in the following were only made possible by the shared effort of all the team members, that I had the pleasure to work with; most importantly Vincent Josse (our team leader), Alain Bernard, Kilian Müller (the PhD students) and Patrick Cheinet (the PostDoc).
At the beginning of my PhD, we substantially improved our setup of a guided atom laser [Bernard 11]. The first goal was to further study 1D AL by making measurements analogous to conductance measurements in condensed matter physics. With the improved apparatus, such studies seemed possible, but we saw a higher possible impact by an in- vestigation of AL in three dimensions (3D) and in the following focused our work on this new subject.
To investigate 3D AL, we transposed the successful scheme that allowed for the obser- vation of one-dimensional Anderson Localization to the three dimensional case. Therefore, we designed a new magnetic levitation and created a very fine-grained disorder by the co-
0 s 2 s 4 s 6 s
Initial Bose Einstein Condensate Localized part (emerging at long time)
among a diffusive part
Time evolution ( disorder + magnetic levitation to cancel gravity)
Figure 1.2: Evidence of 3D localization (from [Jendrzejewski 12a]). By monitoring (via in-situ fluorescence imaging) the time evolution of the atomic density in a strong disorder, we observe the emergence of a localized fraction among a slowly diffusing part.
herent superposition of two laser speckle fields. At the point where we were finally able to start our experiments on Anderson Localization in 3D, Alain and Patrick sadly had to leave our team on their chase for new challenges. Luckily the newest team member Kilian Müller arrived in this very exciting period. In these experiments, we monitored the three-dimensional expansion of an initial BEC in the presence of a quasi-isotropic laser speckle disorder and observed an atomic cloud composed of two components: a localized and a diffusive part (see Fig. 1.2). The comparison of those experiments to theoretical predictions led to a close and very fruitful collaboration with our theory team of Marie Piraud, Luca Pezzé and Laurent Sanchez-Palencia.
Clear as those experiments on AL in 1D and 3D were, none of these experiments includes a direct evidence of the role of quantum interferences. Interestingly, a first order manifestation of coherence even in a weak disorder is observable. The interference between multiple scattering paths leads to the phenomenon of coherent backscattering (CBS), i.e. the enhancement of the scattering probability in the backward direction, due to a quantum interference of amplitudes associated with two opposite multiple scattering paths [Watson 69, Tsang 84, Akkermans 86, Langer 66, Gor’kov 79, Abrahams 79].
In the last part of my PhD we worked on the direct observation of such a CBS peak, which is a direct signal of the role of quantum coherence in quantum transport in disordered media. A cloud of non-interacting ultra-cold atoms was launched with a narrow velocity distribution in an elongated laser speckle disordered potential. Time of flight imaging, after propagation time tin the disorder, directly yield the momentum distribution shown in Fig. 1.3. The most important feature for us is the large visibility peak, which builds up in the backward direction, as it corresponds to the CBS signal.
Outline of the manuscript
• Chapter 2: In this introductory chapter, we start the discussion of the coherent propagation of waves, i.e. ultracold atoms, in disordered potentials. In this chapter
t = 2.5 ms t = 1.5 ms
t = 0 ms 0 pi
Coherent Backscattering Peak
Intial BEC
Figure 1.3: Coherent backscattering(from [Jendrzejewski 12b]). We monitor the time evolution of the momentum distribution (via time of flight imaging) of a BEC launched initially with a narrow velocity distribution. We observe a ring that corresponds to a redistribution of the momentum directions, due to the scattering on a conservative disordered potential. In the backwards directions a large visibility peak, corresponding to the coherent backscattering signal, builds up.
we intend to show the fundamental importance of interference phenomena on the transport properties of the system, as it can lead to the suppression of transport known as Anderson Localization. We further provide the larger overview of exper- imental achievements on AL in the different fields of physics, from classical waves over condensed matter to ultracold atoms.
• Chapter 3: In this second chapter we present the apparatus, which we constructed in order to perform experiments on Anderson Localization in 3D. We give an overview over the experimental techniques used to produce and to control a cloud of ultracold atoms. We discuss the magnetic levitation and the evaporation in the dipole trap, which were finished in this PhD, in some detail.
• Chapter 4: At this point we characterize the properties of the disordered light potential applied to the atoms. It is provided by a speckle potential created by the diffraction of a laser on a rough surface. We show that this disorder is particularly well-controlled and understood as we can completely describe it by its well know cor- relation length and disorder amplitude. Finally, we discuss how a three-dimensional fine-grained disorder is created by the coherent superposition of two perpendicular speckle fields.
• Chapter 5: Having presented the essential ingredients of our experiments, we turn our attention to the most important notions which are necessary for the discussion of our experiments on 3D AL. In the first part we introduce a phenomenological description of the disorder strength. The discussion of the celebrated scaling theory allows us to show that all states are localized in infinite 1D and 2D systems. Further it demonstrates that there is a mobility edge in 3D between the diffusive high energy states and the localized low energy states, which are embedded in the strong disorder.
In the second part of this chapter, we discuss the microscopic quantities governing the quantum transport of ultracold atoms.
• Chapter 6: At this point we have introduced all the tools needed for the discussion
of our expansion experiments on 3D Localization. In a first step, we point out the key points of those experiments in 1D and how we transposed them into our 3D set-up. We then discuss throughout the rest of the chapter in depth our experiments on 3D Localization.
• Chapter 7: We end this manuscript with our findings on Coherent Backscattering of ultracold atoms. After a quick introduction, we explain how we can extract from the evolution of the momentum distribution of the cloud in the disorder important transport quantities, like the scattering time and the transport time. We analyze the time evolution of the Coherent Backscattering signal and end the chapter with first results on the energy dependance of the different transport quantities.
Wave propagation in disorder
Quantum transport differs from classical transport by the crucial role of coherence effects.
In the case of disordered media, it can lead to the complete cancelling of transport when the disorder is strong enough: this is the celebrated Anderson localization (AL) [Anderson 58].
We briefly discuss how interferences can lead to weak localization, which is a precursor of AL, and give a short review of some important experiments on the subject. We then discuss some general properties of AL and the state-of-the-art of the experimental findings on this fascinating and rapidly evolving subject.
2.1 Interference and disorder
Whenever a monochromatic wave is diffracted, interference patterns arise. For example, an interference pattern behind a circular aperture exhibits a set of concentric rings, al- ternating between bright and dark, resulting from constructive or destructive interference (see Fig. 2.1 a). According to Huygens’ principle, this interference picture is obtained from the coherent sum over all possible paths through the aperture. The phase associated with each amplitude is proportional to the path length of the scattered wave divided by its wavelength λ.
a) b)
Figure 2.1: Diffraction patterns. a) Diffraction pattern of a monochromatic wave on a circular hole. (Image from [Akkermans 07]) b) The diffraction of a monochromatic wave on a thick disorder diffuser yields the presented speckle pattern.
Let us now turn to the diffraction of a coherent source by a thin obstacle whose density
fluctuates in space on a scale comparable to the wavelength of the incoming wave. The resulting interference pattern consists of a random distribution of bright and dark areas, as seen in Figure 2.1 b), called a speckle. As long as the obstacle is thin enough, a wave scatters only once in the random medium on its way to the screen. In a thick medium on the other hand, the wave undergoes multiple scattering. Even after this multiple scattering the phase associated to each path is well determined, coherence preserved and the speckle pattern persists.
This interference pattern might be destroyed in different ways. For an incoherent source, the point sources are out of phases for distances larger than the coherence length Lφ. When we illuminate the aperture by an incoherent source of coherence length much smaller than the size of the aperture, the diffraction patterns of the point sources have to be summed incoherently and the interference pattern on the screen disappears. On the other hand, it is possible to employ a coherent light source and rapidly move the obstacle in its plane in a random fashion. In this case, it is the persistence of the detector that averages over many different displaced diffraction patterns and the interference pattern disappeares. These are examples of how interference effects may vanish as the information about the phase gets washed out upon averaging.
2.1.1 Interferences in electron transport in disordered media
The previous idea of wave propagation through a disordered medium also applies to the transport of electrons in metals. In this case, the impurities in the metal are analogous to the scatterers in the optical medium, and the quantity analogous to the intensity is the electrical conductivity. In most cases transport has to be studied for macroscopic samples, with a size bigger than the coherence length Lφ, where one expects interference to be washed out. This makes it possible to treat physical phenomena, such as electrical or thermal transport, employing an essentially classical approach, where we neglect the interference between different paths. The enormous success of this classical approach, developed by Drude, led to the belief that coherent effects would be of no importance for the understanding of transport experiments.
In the 1980s a series of experiments allowed to test these coherence properties for the transport of electrons in more detail. Webb et al. observed the influence of a homogenous magnetic field on the conductance of a very fine metal ring [Webb 85]. Changing the strength of the field changes the relative phase between the top and bottom path through the ring(see picture 2.2 a). The shift of this relative phase and the resulting modified inter- ference pattern manifests itself in oscillations of the conductivity . This is the celebrated Aharonov-Bohm effect [Aharonov 59].
One can also perform the experiment with a cylinder that is much longer than the coherence length, instead of a thin wire. The measured signal after this cylinder can then be interpreted as an average over an ensemble of identical, uncorrelated rings from the experiments by Webb. The experiments by Sharvin and Sharvin, performed in this configuration, showed (Fig. 2.2 b) that even after this ensemble average the observed signal was not flat, but oscillating with a period half as long as in the previous experiments. 1
1It is interesting that things went historically the inverse way. In 1981, the observations by Sharvin and Sharvin were no surprise to the experimentalists and "only" confirmed theoretical predictions of the weak localization phenomena. In 1985, Webb et al. even argued that a merit of their experiment was to not observe the effect of this weak localization.
a) b)
Figure 2.2: Phase coherence effects in disordered conductors. a) The experiments by Webb et al. on the conductance oscillations in a thin ring [Webb 85]. The oscillation period as a function of the flux in the ring is h/e. b) The experiments by Sharvin and Sharvin in a very long cylinder. Oscillations in conductance through this macroscopic sample persist, but their frequency has halve to h/2e[Sharvin 81].
Those experiments have shown that quantum interference effects due to disorder can have important manifestations even on a macroscopic scale after ensemble averaging.
2.2 From weak localization...
To understand the nature of these coherence effects, we will consider the propagation of a wave in a disordered medium, modeled by an random ensemble of point scatterers. We
r
1r
2k
ik
ff(r
1,r
2)
r
1r
2aj aj+1
aj-1
a) b)
Figure 2.3: Propagation in the disorder. a) The initial wave ki enters the disorder at the pointr1. It travels to the endpointr2 with some amplitudef(r1,r2) and leaves the disorder with a wave vector kf. b) Schematic representation of the amplitude f(r1,r2):The propagation from the initial point to the end point can be done on one of the trajectoriesaj. On each of them the wave accumulates a phase δj.
are then interested in the amplitude corresponding to the diffraction from the initial plane wave ki into another wavekf after the propagation in the medium (as shown in Fig. 2.3 a). It may be written in the form
A(kf,ki) =Ø
r1,2
f(r1,r2)ei(ki·r1−kf·r2) , (2.1) wheref(r1,r2) is the amplitude associated with the propagation from r1 tor2 (Fig 2.3 b) and without loss of generality ∆φ= (ki·r1−kf ·r2) is the phase difference between the incoming and outgoing wave. The wave propagates between these endpoints on various trajectories. Each of these trajectories has a certain amplitude |aj| and an associated phase δj. We can then expressf(r1,r2) as a sum of those different paths
f(r1,r2) =Ø
j
|aj|eiδj (2.2)
Finally, the probability associated to A(kf,ki) is given by
P(kf,ki) = |A(kf,ki)|2 (2.3)
= Ø
r3,4
Ø
r1,2
f(r1,r2)f∗(r3,r4)ei(ki·r1−kf·r2)ei(kf·r4−ki·r3) (2.4) We now have to apply the ensemble average, noted by . . ., over different disorder realizations to Eq. (2.4). We note that most of the terms in the productf(r1,r2)f∗(r3,r4) average to zero due to the random phase accumulated on the different trajectories. The only contributing terms are those for which the relative phases vanish. This can only occur for pairs of trajectories, with exactly the same sequence of scattering events, either in the same or in the opposite direction (Fig. 2.4).
r
1r
2k
ik
fr
1r
2k
ik
fa) b)
Figure 2.4: Trajectories with vanishing relative phase. a) Both trajectories are travelled through in the same direction. This contribution is the classical intensity. b) The trajectories are passed through in opposite directions. This term is at the origin of the weak localization and the coherent backscattering effect.
This leaves only two contributions to the ensemble-averaged probability. For the co- propagating path it implies directly that r1 = r3 and r2 = r4, and for the counter- propagating paths we find r1 = r4 and r2 = r3. Hence, the average probability is given by
P(kf,ki) =Ø
r1,2
|f(r1,r2)|2(1 +ei(kf+ki)·(r1−r2)) , (2.5)
where the ∆φ = (kf +ki)·(r1 −r2) in the second term is the phase difference of the counter-propagating trajectories. This phase factor depends on the scattering positions r1,2 and vanishes in general upon ensemble averaging. There are two very interesting and important exceptions to this.
a)
kib)
ki kf
kf
Figure 2.5: The weak localization and coherent backscattering trajectories. a) The CBS peak appears in the backwards direction. b) The weak localization correction.
• Coherent Backscattering (CBS): Constructive interferences persist, if
kf =−ki . (2.6)
This means that the probability of scattering in the direction opposite to the incident one, is doubled to any other direction (Fig.2.5 a).2 This phenomena is known as coherent backscattering is discussed in detail in Ch. 7
• Weak Localization (WL): The other very interesting exception are the terms for which
r1=r2 . (2.7)
They correspond to closed multiple scattering trajectories as shown in Fig. 2.5 b).
They probability of return to the origin is therefore enhanced by these quantum interferences. This increased return probability will diminish the ability of the wave to travel through the disorder and diminish the conductivity, a phenomena known as weak localization.
2.2.1 Experiments on weak localization and backscattering
The discussed mechanism of weak localization and coherent backscattering was initially proposed by several theoretical groups. Interestingly, those ideas were developed in par- allel in the distinct physical fields we have discussed; in optics [Watson 69, de Wolf 71, Barabanenkov 73, Tsang 84, Akkermans 86] and at the same time in the context of con- densed matter physics [Langer 66, Gor’kov 79, Abrahams 79]. In condensed matter, the weak localization phenomenon (see e.g. [Altshuler 82, Akkermans 07]) is responsible for the anomalous resistance of thin metallic films [Dolan 79, Bishop 80, Van Den Dries 81]
2Actually, the CBS has a finite width as we will explain in Ch. 7.
and its variation with an applied magnetic field [Bergmann 84]. For example, studies on the magnetoresistance of thin Mg films showed convincing proof of weak localization (Fig.
2.6 a). In such experiments the applied magnetic field dephases the counter-propagating paths. This diminishes the effect of WL and leads to the observed decrease of the resis- tance. It seems very difficult to directly observe CBS of electrons and we are not aware any experiments on this.
a) b)
R.lOOii
¶aMa
Figure 2.6: Experiments on WL and CBS. a) Measurements of the magnetoresistance of Mg films [Bergmann 84]. The magnetic field dephases the loops and diminishes the effect of the WL, which leads to the observed decrease of the resistance. b) The backscattered intensity is observed and yields an enhancement factor of 2[Wiersma 95].
The continuous black line is the theoretical prediction for the shape of the CBS peak [Akkermans 86].
As WL is ubiquitous to wave physics the theoretical predictions triggered considerable experimental efforts to observe further direct signatures of weak localization in classi- cal waves. Shortly thereafter, coherent backscattering was observed in optics [Kuga 84, van Albada 85, Wolf 85]. Still, it took another ten years after the first observation of coherent backscattering until the doubling of the backscattered amplitude was observed by Wiersma et al. in 1995 [Wiersma 95] (see Fig. 2.6 b). In the following, CBS of light has been observed in a numerous situations of multiple scattering: in suspensions of liquid diffusers [Kuga 84, Wolf 85], on cold atoms [Labeyrie 99], with speckle fields [van Tiggelen 90], and even with incoherent light like the light from the sun [Okamoto 96, Lenke 97]. Nowadays, the study of CBS was extended to other systems of classical waves like acoustics [Bayer 93, Tourin 97] and seismology [Larose 04].
2.3 ... to the suppression of transport - Anderson Localiza- tion
The presented theoretical and experimental studies are the first order manifestation of interference effects on quantum (or wave) transport through disorder media. They clearly showed that quantum transport differs from classical transport by the crucial role of
coherence effects. They can even lead to the complete suppression of transport when the disorder is strong enough: the famous Anderson localization [Anderson 58]. In this localized regime, the propagation is inhibited and the wave (or quantum particle) remains localized around its initial position. Anderson showed further that the localized wave function has a spatial exponential decay from the point of localization with a localization lengthξ.
As so often in physics, the historical development of localization did not follow the logics of how localization is often presented nowadays. Initially predicted by P.W. Anderson for spins or electrons in "certain random lattices" in 1958, the importance of AL was underestimated for a long time. This was probably due to the complexity of the paper and the absence of a clear physical mechanism giving rise to localization.3
For some years, it was Mott4 and coworkers, who sustained the study of Anderson Localization. They gave a qualitative criterion of when the localization should arise.
As the influence of the disordered medium is increased, the approach outlined in the previous section breaks down when the wavelength λdB of the incident wave becomes of the same order of magnitude as the mean free path l, the typical distance between successive scattering events. Then the particle is continuously scattered by the disorder and can not propagate anymore. Effectively, a transition from the diffusive regime to the strongly or Anderson localized regime, where the transport stops, takes places. The position of this transition, calledmobility edge, is given by the Ioffe-Regel criterionkl≈1 [Ioffe 60].
Only in the 70s and 80s, twenty years after Anderson’s pioneering paper, the interest in localization increased considerably and our current picture of localization emerged. At this point, people understood that localization should highly depend on the dimension- ality of the system. Initial pioneering works by Langer [Langer 66], Wegner [Wegner 76], Thouless [Thouless 77], and Gorkov [Gor’kov 79] layed then the ground for the scaling the- ory by Abrahams, Anderson, Licciardello and Ramakrishnan [Abrahams 79]; the so-called
"gang of four". From the scaling theory the importance of interferences for localization became clearer [Lee 85] and the previously presented experimental investigations on weak localization started. Further, it was shown in this context that all states are localized in a one dimensional systems even in the presence of very weak disorder [Landauer 70]. This is also the case in 2D, although here the localization length increases exponentially when the influence of disorder decreases.5 In 3D, the behavior is quite different: there is a real phase transition between a diffusive regime and a localization regime. Here the critical exponents (sand ν) characterizing the transition can be defined in part by the evolution of the characteristic parameters of the system around the transition, namely the diffusion constant and the localization length:
D ∝ |E−EC|s forE ≥EC (2.8)
ξ ∝ |EC−E|−ν forE ≤EC , (2.9)
where E is the wave energy and EC the critical energy corresponding to the transi- tion. In the community there is now general agreement, based on numerical simulations,
3An excellent review on AL in general and its historical evolution can be found in the book "50 years of Anderson Localization" edited by E. Abrahams [Anderson 10].
4The father of the celebrated Mott transition.
5The possible existence of a mobility edge in 2D, strongly supported by Mott, had been subject of a long going debate before.
[Kramer 93] that both exponents are s=ν = 1.58. But even 50 years after the pioneering work of Anderson, there is still no analytical prediction for the position of the mobility edge and an experimental determination of the critical exponents remains a major challenge.
2.3.1 Experiments on Anderson Localization
Anderson localization has of course been the subject of numerous experimental studies [Lee 85, Kramer 93]. Originally predicted in the context of condensed matter, the first studies have been conducted in electronic systems. The study of the consequences of weak localization were therefore extended to the ones of strong localization regime. Measure- ments of the conductivity and dielectric susceptibility of 2D disordered semiconductors, silicon doped [Rosenbaum 83] and AlxGa1−xAs [Katsumoto 87], allowed the first obser- vations of the metal-insulator transition in the 80’s. However, interactions between elec- trons and the presence of thermal excitations made quantitative measurements in such electronic systems very delicate and better controlled systems were needed. One actually had to wait until 1997 to have experimental evidence for Localization in 1D electronic conductors [Gershenson 97].
a)
b) c)
!!"
yyy
Figure 2.7: Anderson localization of classical waves. Anderson Localization of clas- sical waves has been observed, for example, with a) photonic crystals [Schwartz 07] b) ultrasound [Hu 08] and c) light [Wiersma 97].
Since the 90s, Anderson Localization is therefore also studied outside of the field of condensed matter in other systems, which are easier to monitor (see Fig. 2.7). Being linked to the presence of interferences, AL is observable in all wave systems: as well clas- sical as quantum. In particular, the previously mentioned observations of CBS in the 1980s and a proposal of S. John [John 88] in 1988 have stimulated intense research around Anderson Localization of acoustic or electromagnetic waves, such as µ-waves [Hauser 92]
and light waves [Wiersma 97] (see Fig. 2.7). Although these systems are a priori more easily controlled, especially because of the absence of interactions, the obtained results
were controversial. In such experiments it is difficult to differentiate between the signa- tures of localization of those of absorption: both lead to an exponential decrease of the wave function as a function of the thickness of the scattering medium. However, this problem was overcome by measuring the fluctuations of the transmission [Chabanov 00]
or by studying its dynamics [Chabanov 03, Störzer 06, Lagendijk 09]. Finally, recent ex- periments in disordered photonic crystals [Schwartz 07, Lahini 08], µ-waves [Laurent 07]
and ultrasound [Hu 08] have directly observed the localized wave functions.
2.4 Experiments with ultracold atoms
Only relatively recently cold atoms have been proposed to study the problem of localiza- tion [Damski 03]. Ultracold atoms have proven to be a formidable system for the study of condensed matter physics [Bloch 08]. They allow to implement systems in any dimension;
the control of the interatomic interactions, either by density control or by Feshbach reso- nances, the possibility to design well-controlled and phonon-free disordered potentials, the application of synthetic magnetic fields [Lin 09], and the opportunity to measure in situ atomic density profiles via direct imaging.
z (mm) 1 10 100
–0.8 –0.4 0.0 0.4 0.8
z
t = 0
t > 0
a) b) c)
Figure 2.8: 1D systems showing Anderson localization of ultracold atoms. First observation on 1D localization of ultracold atoms by a) the Aspect group [Billy 08] b) the Inguscio group [Roati 08]. c) The Anderson transition was investigated using the kicked rotor systems in the Garreau group [Chabé 08].
The first experiments, conducted in 2005 [Schulte 05, Fort 05, Clément 05], with a disorder created by a optical speckle pattern, however, were inconclusive, partly because of too strong interatomic interactions. Theoretical work that followed [Sanchez-Palencia 07, Fallani 08, Sanchez-Palencia 08] provided a better understanding of the regime which was necessary to reach. In 2008, Anderson localization was finally observed in one dimension (Fig. 2.8) by our team [Billy 08]. In parallel, the Inguscio group in Florence had meanwhile highlighted the phenomenon in a one-dimensional bichromatic lattice, reproducing the Aubry-André model, which exhibits a transition to a localized regime [Roati 08].
Cold atoms have also been used to demonstrate Anderson Localization in momentum space for kicked rotor systems. The Hamiltonian of these systems can be mapped on the Hamiltonian describing the Anderson localization, by replacing the direct space co- ordinates by coordinates in momentum space [Casati 89]. With these systems it is also possible to explore with Anderson Localization in different dimensions by adding kicks at
incommensurable frequencies. The experiments were performed in 1D [Moore 94], and 3D in the team of Jean-Claude Garreau in Lille [Chabé 08, Lemarié 10, Lopez 12].
2.5 Conclusion - Towards higher dimensions and interacting bosons
The success of the experiments in 2008 proved the utility of ultracold atoms to investi- gate disordered systems. It was then the logical next step to use these systems for the investigation of more complex systems. These efforts focus nowadays on two important properties: the influence of dimensionality and interactions.
With interactions the simple wave picture breaks down rapidly and the problem gets extremely difficult to treat. A particular interest has been put on the "dirty bosons"
problem, which is still a major challenge for theory and experiment, see i.e. [Lugan 07, Paul 07, Falco 09a] and references therein. Especially the conditions for the appearance of an insulating phase called Bose glass [Giamarchi 88, Fisher 89] remain controversial [Roux 08, Roscilde 08, Pollet 09, Gurarie 09]. Recently the first experimental results for disordered, weakly interacting bosons in 1D were published by the groups of Inguscio [Deissler 10], and of Randall Hulet in Houston [Dries 10]. First experiments on the in- fluence of disorder on the BKT transition in 2D were presented by the Krub experi- ment in our group [Allard 12] and the Rolston group in Washington [Beeler 12]. The observed shift of the critical temperature awaits now to be confronted to different theories [Lopatin 02, Pilati 08, Pilati 10]. Meanwhile, the group of Brian DeMarco at Urbana has developed an experimental system to achieve the disordered Bose-Hubbard model, and thus to probe the regime of strong interactions [White 09, Pasienski 10].
In the non-interacting case, it is extremely interesting to investigate higher dimensions and target the investigation of the critical region around the mobility edge. First results on
diffusion in 2D where obtained by our the 2D experiment of our group [Robert-de Saint-Vincent 10].
With the kicked rotor system refined measurements of the critical exponents were per- formed. The measured value of ν = 1.63 ±0.05 is very close to the expected value [Kramer 93]. To our knowledge, this is the only experimental observation of the critical exponents in agreement with numerical predictions up to today. The precise determination of these critical exponents in a disorder is therefore still an issue for which cold atoms could be used [Sanchez-Palencia 10], and is one of the objectives of the experiment described in this manuscript. In 2012, two groups published on the observation of 3DLocalization of ultracold atoms. We demonstrated evidence for localization of bosons in a speckle disorder [Jendrzejewski 12a], while the deMarco group reported on studies on the expansion of a Fermi gas in a very anisotropic disorder [Kondov 11].
The main part of this manuscript is then devoted to our efforts leading towards a precise investigations of the critical region. We discuss general important notions on Localization in more detail in Ch. 5 and our experimental findings on Anderson Localization of ultracold atoms in three dimensions in Ch. 6. Convincing as existing experiments on AL with ultracold atoms are, they do no provide a direct proof of coherence. We will present in the last chapter of this manuscript our results on CBS of ultracold atoms [Jendrzejewski 12b], see Ch. 7, obtained simultaneously with the Labeyrie group in Nice [Labeyrie 12]. Those experiments provide a first direct observation of phase coherent transport with ultracold atoms.
Production of coherent matter waves
Bose-Einstein condensates (BEC) of dilute atomic clouds provide unique opportunities for exploring quantum phenomena on a macroscopic scale. They have first been realized in 1995 in the Boulder group with rubidium atoms [Anderson 95] and only months later in the Ketterle group at the MIT with sodium atoms [Davis 95a]. In these experiments a dilute cloud is cooled to temperatures in the nano-kelvin regime. At a critical temperature the wave packets of the different bosonic atoms overlap and a Bose-Einstein condensate develops. In such a condensate the lowest energy state gets collectively occupied by a macroscopic number of atoms. With our experimental apparatus we prepare condensate with several 104 atoms at some nK as a starting points for our subsequent experiments with ultracold atoms in a disordered speckle potential.
We begin this chapter with a short introduction to the theoretical description of Bose- Einstein condensates. We then describe the experimental set-up for the production of the Bose-Einstein condensate.
3.1 Generalities about Bose-Einstein condensates
In this section we will discuss some important theoretical results about Bose-Einstein condensates that we will use in the following to describe our experiments. We will only present some major results here and leave out their derivation. For more details we refer the reader to the detailled reviews on Bose-Einstein condensates [Dalfovo 99, Ketterle 99, Castin 01, Leggett 01].
3.1.1 Bose-Einstein condensation and the Gross-Pitaevskii Equation Bose-Einstein condensation [Dalfovo 99] corresponds the the accumulation of a macro- scopic number of bosons, integer-spin particles, in the lowest energy state of the quantum system. Qualitatively, it occurs when the wave packets describing the individual bosons start to overlap. In other words, the atoms condense in the fundamental state once the de Broglie wavelength λdB is at the order of magnitude or bigger than the inter-particle distance d∼n−1/3.
D=nλ3dB ∼1 withλdB =
ó 2π~2
mkBT , (3.1)
where Dis the phase-space density,m the mass of the atoms,T their temperature and n is the density of the cloud. In the experiment we work with a three-dimensional harmonic trap with curvatures ωx,y,z. The phase-space-density in such a harmonic trap is given by
[Dalfovo 99]
D= 3~ω
kB 43 N
T3 , (3.2)
where ¯ω = (ωzωyωz)1/3 and N is the total number of atoms. Then, the condition for condensation Eq. (3.1) can be translated into a critical temperature
Tc = 0.94~ω¯
kBN1/3 . (3.3)
Below the transition temperature the atoms accumulate in the ground state of the trap and clouds with high atomic densities are obtained. Because of these high densities, in- teractions via two-body collisions are very frequent and have to be taken into account for the description of the ground state. These collisions are well characterized by the scattering length a, which describes the effective size of the atoms for s-wave scatter- ing [Chikkatur 00]1. In our experiments we use the repulsively interacting 87Rb with a scattering length of a= 5.3nm.
Because we work with very dilute gases (na3 ¹1) , we can use a mean-field description for the interactionsVint(r) =g·n(r) with the coupling constant g= 4πm~2a. The evolution of the wave function ψ(r, t) of such a weakly-interacting condensate with N atoms in an external potential V(r) is driven by the time-dependent Gross-Pitaevskii Equation
i~∂
∂tψ(r, t) = C
−~2
2m∆ +V(r) +g|ψ(r, t)|2 D
ψ(r, t) . (3.4) In the stationary regime, we can write the wave function as ψ(r, t) = φ0(r)e−iµ/~t.
|φ0(r)|2 = n(r) is the spatial density and µ the chemical potential of the condensate, which describes the increase of the total energy by adding an atom to the condensate with N atoms. The leads to the stationary Gross-Pitaevskii equation
C
−~2
2m∆ +V(r) +g|φ0(r)|2 D
φ0(r) =µφ0 . (3.5)
The first term on the left-hand side of the equation describes the kinetic energy of the condensate, the second one its potential energy and the third one the interaction energy.
3.1.2 The Thomas-Fermi Regime
We can estimate the kinetic and interaction energy of a condensate in a harmonic trap using the size of the harmonic oscillator σ =ð~/mω
Ekin ∼ N ~2
2mσ2 (3.6)
Eint = N2g 2
3
4πσ3 . (3.7)
1Scattering in higher order orbitals likedandg-wave are negligible in our experiments due to the very low energies we are working at [Thomas 04, Buggle 04]
When the number of atoms is large N º σ/a, we can neglect the kinetic energy in comparison to the interaction energy. In this Thomas-Fermi regime, the spatial density takes the simple form
n(r) = max
3µ−V(r) g ,0
4
(3.8) ntrap(x, y, z) = max
µ/g− Ø
i=x,y,z
mωi2 2g ri2,0
, (3.9)
where the second quantity describes the profile in a harmonic trap. It is an inverted parabola with radius
rT F,i= ó 2µ
mω2i , (3.10)
where i=x, y, z indicates the direction. Via the normalization of n(r) we determine the chemical potential
µ= 1 2
115aN~2ω¯322/5m1/5 . (3.11) We can rewrite the condition for the validity of the Thomas-Fermi approximation as µ º ~ωi. In our experiment we have typical chemical potentials of µ/h ∼ 50Hz with trapping frequencies of ωi/2π ∼5Hz. Hence, our trapped condensates are in the Thomas- Fermi regime, where we can neglect the kinetic energy and their interaction energy is given by [Dalfovo 99]
Eint = g 2
Ú
drn(r)2 (3.12)
⇒Eint,T F = 2
7µ. (3.13)
The most common way to characterize those Bose-Einstein-condensates are the time-of- flight experiments we are going to present in the following.
3.1.3 Expansion of a BEC: time-of-flight experiments
The experiments presented in this manuscript use the BEC as a source of matter waves to study their transport properties. The easiest way to let these matter waves propagate, is to cut the trap suddenly and let the atoms expand freely. During this expansion the interaction energy Eintis transformed into kinetic energyEkin. We will use this technique in order to create a cloud of non-interacting atoms with a certain velocity distribution as described in the following.
At the beginning of the expansion the kinetic energy is negligible and the only contri- bution to the total energy, called the release energy, is the interaction energy
Erel =Eint,T F = 2
7µ . (3.14)
After a time of several 1/ωi, the interaction energy is transformed into kinetic energy. We thus have created a non-interacting wave packet with a typical energy spread on the order of µ. Given the simplicity of these so called time-of-flight (TOF) experiments, they are a commonly used tool to characterize the properties of the condensate.
The dynamics of the condensate can be studied in the framework of the time-dependent Gross-Pitaevskii equation 3.4. In the approximation of the Thomas-Fermi regime it can be shown that the condensate is following some scaling laws that we are going to discuss now.
3.1.3.1 The scaling approach
It was shown that the expansion of a BEC is described by a rescaling of the initial parabolic shape of the cloud [Kagan 96, Castin 96]. Hence, the atomic density in the direction i evolves over time with
n(ri, t) = 1
bx(t)by(t)bz(t) ·n(ri/bi(t),0) (3.15) with d2bi(t)
dt2 = ω2i
bibxbybz −ωres,i2 bi(t) , (3.16) whereωres,i=x,y,z(t) are residual frequencies after the switch-off2. Thebi(t) are the rescal- ing functions with initial conditions bi(0) = 1 and ˙bi(0) = 0.
For very long expansion times, where the size of the expanding cloud is much bigger than the initial size, the observed profilen(r, t) gives the velocity distribution of the atoms by n(vi =ri/t) 3. Thus, we can calculate the velocity distribution from Eq. 3.16
n(vx, vy, vz, t) = max A15N
8π¯v3 A
1−Ø
i
vi2 v2i,max(t)
B ,0
B
, (3.17)
where ¯v= (vx,max(t)vy,max(t)vz,max(t))1/3 is the average speed andvi,max(t) = ˙bi(t)·rT F,i is the maximum velocity.
In general, the maximum velocity has to be determined from the equations (3.16). The analytical solution of these equations are only known in several limiting cases.
3.1.3.2 Ballistic expansion
A common situation is the free expansion, where the residual frequencies are negligible ωi,res= 0.4
From Eq. (3.16) we see that the typical time scale of the evolution is 1/ωi. After this time the released energy is given by the total kinetic energy.
Erel(t→ ∞) = Ekin,tot(t→ ∞) (3.18)
= Ú
dv n(v, t→ ∞)·m
2v2 (3.19)
Using Eq. (3.17) and (3.14), this leads to the relation Ekin,max= m
2 Ø
i
v2i,max(t→ ∞) = 2µ (3.20)
2In our case coming from the magnetic levitation, see Sec. 3.4.3
3This is not the velocity distribution in the trap. The interaction energy has now been converted into kinetic energy
4An analytical solution was derived for the case of the release from a very elongated trap in [Castin 96]
.
3.1.3.3 Expansion from an isotropic trap
In our experiment we create a BEC in a quasi-isotropic trap with trapping frequency ωiÄωtrap and negligible residual trapping after the switch-offωres,iÄ0. In this case, we can deduce vmax directly from Eq. (3.20)
mvmax2 (t→ ∞)
2 = 2µ
3 (3.21)
vmax2 = 4µ
3m (3.22)
This relation allows us in the experiments to measure precisely the chemical potential in the trap from TOF-experiments as we will discuss in Sec. 3.4.4.
3.2 Manipulation of the atoms
In the previous section we introduced the most important notions on Bose-Einstein con- densates. In this section, we want to describe how we can manipulate neutral atoms with magnetic and optical interactions in order to achieve such Bose-Einstein condensates.
3.2.1 Magnetic potentials
The magnetic momentumµof the atoms is coupled to a static magnetic fieldB(r) by the potential Vmag(r) =−µ·B(r). The energy shift induced by this coupling is described by the Breit-Rabi formula for 87Rbin the 52S1/2 ground-state [Breit 31]:
Emag,± =±h∆ν 2
ñ
1 +mFξ+ξ2 withξ= 2µB|B(r)|
h∆ν , (3.23)
where ∆ν = 6.835GHz is the splitting between the hyperfine levelsF = 1 andF = 2. mF
is the quantum number which corresponds to the projection off the magnetic moment on the local direction of B(r). In this Breit-Rabi formula the potential energy depends only on the norm of the magnetic field and not its direction, because the spin of the atoms follows the magnetic field lines adiabatically with the Larmor frequency.
We have pictured the potential energy as a function of the strength of the magnetic field in Fig. 3.1. For weak magnetic fields the induced Zeeman shift is proportional to the norm of the magnetic field|B(r)|:
Vmag,Z(r) =gFµBmF|B(r)| (3.24) where gF is called the Lande factor (gF=1 = −1/2 and gF=2 = 1/2). If the field is strong the interaction term dominates the hyperfine energies. In this case, the slope is proportional to the quantum number associated to the fine structuremJ.
3.2.2 Optical potentials
If the atom is interacting with a laser it will experience forces of two different kinds [Cohen-Tannoudji 98b]. We can mostly understand them in terms of a two-level atom with a transition frequency ω0/2π and lifetime Γ−1 of the excited state. We note the
